Suppose ∑ n = 0 ∞ a n z n has radius of convergence R and σ N ( z ) = | ∑ n = N ∞ a n z n | . Suppose | z 1 | < | z 2 | < R , and T is either z 2 or a neighborhood of z 2 . Put \sigma _N \left( z \right){\text{ for }}z\varepsilon T} \right\}$"> S = { N | σ N ( z 1 ) > σ N ( z )  for  z ϵ T } . Two questions are asked: (a) can S be cofinite? (b) can S be infinite? This paper provides some answers to these questions. The answer to (a) is no, even if T = z 2 . The answer to (b) is no, for T = z 2 if lim a n = a ≠ 0 . Examples show (b) is possible if T = z 2 and for T a neighborhood of z 2 .